LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 52

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – April 2009

S  815 – PROBABILITY THEORY

 

 

 

Date & Time: 04/05/2009 / 1:00 – 4:00       Dept. No.                                            Max. : 100 Marks

 

 

SECTION-A (10 × 2=20)

Answer ALL the questions.

 

1. With reference to tossing a regular coin once and noting the outcome, identify

completely all the elements of the probability space (Ω, A, P).

∞

2. If P(A_n ) =1, n=1,2,3,… , evaluate P( ∩  A_n  ) .

n=1

3. Show that the limit of any convergent sequence of events is an event.
4. Define a random variable and its probability distribution.
5. Calculate E(X), if X has a distribution function F(x), where

F(x) =   0         if x<0

x/2      if o≤ x<1

• if x ≥ 1.

6. If X₁ and X₂ are independent random variables and g₁ and g₂ are Borel functions, show

that g₁(X₁) and g₂(X₂ )are independent.

7. State Glivenko-Cantelli theorem.
8. Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
9. State Kolmogorov’s strong law of large numbers(SLLN).
10. State Lindeberg-Feller central limit theorem.

 

SECTION-B (5 × 8 = 40)

Answer any FIVE questions.

 

11. Define the distribution function of a random variable X. State and establish its

defining properties.

12. Explain the independence of two random variables X and Y. Is it true that if X and

Y are independent, X² and Y² are independent? What about the converse?

13. State and prove Borel zero –one law.
14. State and prove Kolmogorov zero-one law for a sequence of independent random

variables.

15. Define convergence in probability. Show that convergence in probability implies

convergence in distribution.

16. a) Define “Convergence in quadratic mean” for a sequence of random variables.
17. b) X is a random variable, which takes on positive integer values. Define

X_n   = n+1   if X=n

n      if X=(n+1)

X      otherwise

Show that X_n  converges to X  in quadratic mean.      (2+6)

 

17. Establish the following:

(a) If  X_n → X with probability one, show that X_n → X in probability.

(b) Show that X_n → X almost surely if and only if for every є >0,

P [lim sup │ ‌X_n – X│> є ]= 0

18. Let { X_n ,n ≥1} be a sequence of independent random variables such that X_n has

uniform distribution on (-n, n). Examine whether the central limit theorem holds for

the sequence { X_n, n≥1}.

 

SECTION-C (2 x 20 = 40 marks)

Answer any TWO questions

 

19.a) Show that the probability distribution of a random variable is determined by its

distribution function.

1. b) Show that the vector X =(X₁, X_2,…, X_p ) is a random vector if and only if X_j,

j =1, 2, 3… p is a real random variable.

1. c) If X is a random variable with continuous distribution function F, obtain the

probability distribution of F(X).                              (6+8+6)

20.a) State and prove Kolmogorov’s inequality.        (10 marks)

1. b) State and prove Kolmogorov three series theorem for almost sure convergence of

the series Σ X_n of independent random variables.  (10 marks)

21.a) If  X_n and Y_n  are independent for each n, if  X_n →  X,  Y_n → Y, both in

distribution, prove that (X_n² + Y_n²) → (X²+Y²) in distribution.            (10 marks)

1. b) Let { X_n } be a sequence of independent random variables with common frequency

function f(x) = 1/x²  , x=1,2,3,… Show that X_n /n does not converge to zero with

probability one.                                                                          (10 marks)

22.a) State and prove Levy continuity theorem for a sequence of characteristic

functions.(12 marks).

b)Let {X_n} be a sequence of normal variables with E (X_n) = 2 + (1/n) and

var(X_n) = 2 + (1/n²), n=1, 2, 3…Examine whether the sequence converges in

distribution.(8 marks).

 

 

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